Linear Algebra Done Right · Problem 8

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · gram-schmidt-procedure

Problem

Make

\mathcal{P}_2(\mathbf{R})

into an inner product space by defining

\langle p, q \rangle = \int_0^1 pq

for all

p, q \in \mathcal{P}_2(\mathbf{R})

. (a) Apply the Gram--Schmidt procedure to the basis

1, x, x^2

to produce an orthonormal basis of

\mathcal{P}_2(\mathbf{R})

. (b) The differentiation operator (the operator that takes

p

p'

) on

\mathcal{P}_2(\mathbf{R})

has an upper-triangular matrix with respect to the basis

1, x, x^2

, which is not an orthonormal basis. Find the matrix of the differentiation operator on

\mathcal{P}_2(\mathbf{R})

with respect to the orthonormal basis produced in (a) and verify that this matrix is upper triangular, as expected from the proof of 6.37.